Basic invariants
| Dimension: | $4$ |
| Group: | $S_5$ |
| Conductor: | \(7031\)\(\medspace = 79 \cdot 89 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.3.7031.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | odd |
| Determinant: | 1.7031.2t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.3.7031.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{3} - x^{2} - x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 57 a + 50 + \left(35 a + 28\right)\cdot 59 + \left(49 a + 10\right)\cdot 59^{2} + \left(49 a + 53\right)\cdot 59^{3} + \left(17 a + 56\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 + 15\cdot 59 + 46\cdot 59^{2} + 53\cdot 59^{3} + 35\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 a + 47 + \left(52 a + 46\right)\cdot 59 + \left(45 a + 21\right)\cdot 59^{2} + \left(55 a + 3\right)\cdot 59^{3} + \left(57 a + 58\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 48 + \left(23 a + 7\right)\cdot 59 + \left(9 a + 24\right)\cdot 59^{2} + \left(9 a + 53\right)\cdot 59^{3} + \left(41 a + 24\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 38 a + 9 + \left(6 a + 19\right)\cdot 59 + \left(13 a + 15\right)\cdot 59^{2} + \left(3 a + 13\right)\cdot 59^{3} + \left(a + 1\right)\cdot 59^{4} +O(59^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 5 }$ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $10$ | $2$ | $(1,2)$ | $2$ | ✓ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ | |
| $20$ | $3$ | $(1,2,3)$ | $1$ | |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ | |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ | |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |